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Theorem cnextf 21159
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1  |-  C  = 
U. J
cnextf.2  |-  B  = 
U. K
cnextf.3  |-  ( ph  ->  J  e.  Top )
cnextf.4  |-  ( ph  ->  K  e.  Haus )
cnextf.5  |-  ( ph  ->  F : A --> B )
cnextf.a  |-  ( ph  ->  A  C_  C )
cnextf.6  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
cnextf.7  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
Assertion
Ref Expression
cnextf  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, J    x, K    ph, x

Proof of Theorem cnextf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4  |-  ( ph  ->  J  e.  Top )
2 cnextf.4 . . . 4  |-  ( ph  ->  K  e.  Haus )
3 cnextf.5 . . . 4  |-  ( ph  ->  F : A --> B )
4 cnextf.a . . . 4  |-  ( ph  ->  A  C_  C )
5 cnextf.1 . . . . 5  |-  C  = 
U. J
6 cnextf.2 . . . . 5  |-  B  = 
U. K
75, 6cnextfun 21157 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Haus )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Fun  ( ( JCnExt K
) `  F )
)
81, 2, 3, 4, 7syl22anc 1293 . . 3  |-  ( ph  ->  Fun  ( ( JCnExt
K ) `  F
) )
9 simpl 464 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ph )
10 cnextf.6 . . . . . . . . 9  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
1110eleq2d 2534 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
1211biimpar 493 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( ( cls `  J
) `  A )
)
13 cnextf.7 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
14 n0 3732 . . . . . . . 8  |-  ( ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/)  <->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
1513, 14sylib 201 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
16 haustop 20424 . . . . . . . . . . . . . 14  |-  ( K  e.  Haus  ->  K  e. 
Top )
172, 16syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
185, 6cnextfval 21155 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
191, 17, 3, 4, 18syl22anc 1293 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( JCnExt K
) `  F )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2019eleq2d 2534 . . . . . . . . . . 11  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  <. x ,  y
>.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
21 opeliunxp 4891 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2220, 21syl6bb 269 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2322exbidv 1776 . . . . . . . . 9  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
24 19.42v 1842 . . . . . . . . 9  |-  ( E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2523, 24syl6bb 269 . . . . . . . 8  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  ( x  e.  ( ( cls `  J ) `
 A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2625biimpar 493 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) )
279, 12, 15, 26syl12anc 1290 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)
2825simprbda 635 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  ( ( cls `  J
) `  A )
)
2911adantr 472 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
3028, 29mpbid 215 . . . . . 6  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  C )
3127, 30impbida 850 . . . . 5  |-  ( ph  ->  ( x  e.  C  <->  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) ) )
3231abbi2dv 2590 . . . 4  |-  ( ph  ->  C  =  { x  |  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) } )
33 dfdm3 5027 . . . 4  |-  dom  (
( JCnExt K ) `
 F )  =  { x  |  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) }
3432, 33syl6reqr 2524 . . 3  |-  ( ph  ->  dom  ( ( JCnExt
K ) `  F
)  =  C )
35 df-fn 5592 . . 3  |-  ( ( ( JCnExt K ) `
 F )  Fn  C  <->  ( Fun  (
( JCnExt K ) `
 F )  /\  dom  ( ( JCnExt K
) `  F )  =  C ) )
368, 34, 35sylanbrc 677 . 2  |-  ( ph  ->  ( ( JCnExt K
) `  F )  Fn  C )
3719rneqd 5068 . . 3  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  =  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
38 rniun 5252 . . . 4  |-  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  = 
U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
39 vex 3034 . . . . . . . . 9  |-  x  e. 
_V
4039snnz 4081 . . . . . . . 8  |-  { x }  =/=  (/)
41 rnxp 5273 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
4240, 41ax-mp 5 . . . . . . 7  |-  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )
4311biimpa 492 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  x  e.  C )
446toptopon 20025 . . . . . . . . . . 11  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
4517, 44sylib 201 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (TopOn `  B ) )
4645adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  K  e.  (TopOn `  B )
)
475toptopon 20025 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
481, 47sylib 201 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  C ) )
4948adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  J  e.  (TopOn `  C )
)
504adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  A  C_  C )
51 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
52 trnei 20985 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
5352biimpa 492 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
5449, 50, 51, 12, 53syl31anc 1295 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
553adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  F : A --> B )
56 flfelbas 21087 . . . . . . . . . . 11  |-  ( ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  /\  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  -> 
y  e.  B )
5756ex 441 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  ->  y  e.  B ) )
5857ssrdv 3424 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
5946, 54, 55, 58syl3anc 1292 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
6043, 59syldan 478 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
A ) ) `  F )  C_  B
)
6142, 60syl5eqss 3462 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ran  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6261ralrimiva 2809 . . . . 5  |-  ( ph  ->  A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
63 iunss 4310 . . . . 5  |-  ( U_ x  e.  ( ( cls `  J ) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B 
<-> 
A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6462, 63sylibr 217 . . . 4  |-  ( ph  ->  U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6538, 64syl5eqss 3462 . . 3  |-  ( ph  ->  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6637, 65eqsstrd 3452 . 2  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  C_  B )
67 df-f 5593 . 2  |-  ( ( ( JCnExt K ) `
 F ) : C --> B  <->  ( (
( JCnExt K ) `
 F )  Fn  C  /\  ran  (
( JCnExt K ) `
 F )  C_  B ) )
6836, 66, 67sylanbrc 677 1  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756    C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   U_ciun 4269    X. cxp 4837   dom cdm 4839   ran crn 4840   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995   clsccl 20110   neicnei 20190   Hauscha 20401   Filcfil 20938    fLimf cflf 21028  CnExtccnext 21152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-pm 7493  df-rest 15399  df-fbas 19044  df-fg 19045  df-top 19998  df-topon 20000  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-cnext 21153
This theorem is referenced by:  cnextcn  21160  cnextfres1  21161
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